Billiards and Hofer's Geometry

TL;DR

This paper establishes a connection between billiard dynamics in convex domains and Hofer's geometry, providing bounds on the Hofer distance between billiard maps and exploring their symplectic properties.

Contribution

It introduces a novel link between billiard dynamics and symplectic topology, specifically relating billiard maps to Hofer's geometry and the group of area-preserving maps.

Findings

Hofer distance between billiard maps is bounded by geometric distance between tables

Billiard maps of convex polygons are in the Hofer metric completion of area-preserving maps

Connections to dynamics and open problems are discussed

Abstract

We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the annulus. Finally, we discuss related connections to dynamics and pose several open problems.